An Explicit Formula for the Arithmetic–Geometric Mean in Genus 3

Abstract

The arithmetic–geometric mean algorithm for calculating elliptic integrals of the first type was introduced by Gauss. The analogous algorithm for abelian integrals of genus 2 was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for abelian integrals of genus 3.

Keywords:

• Primary: 14H40, 14H45, 14Q05
• Prym varieties
• arithmetic geometric mean